cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A076478 The binary Champernowne sequence: concatenate binary vectors of lengths 1, 2, 3, ... in numerical order.

Original entry on oeis.org

0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

N. J. A. Sloane, Nov 10 2002

Keywords

Comments

Can also be seen as triangle where row n contains all binary vectors of length n+1. - Reinhard Zumkeller, Aug 18 2015
From Clark Kimberling, Jul 18 2021: (Start)
In the following list, W represents the sequence of words w(n) represented by A076478. The list includes five partitions and two self-inverse permutations of the positive integers.
length of w(n): A000523
positions in W of words w(n) such that # 0's = # 1's: A258410;
positions in W of words w(n) such that # 0's < # 1's: A346299;
positions in W of words w(n) such that # 0's > # 1's: A346300;
positions in W of words w(n) that end with 0: A005498;
positions in W of words w(n) that end with 1: A005843;
positions in W of words w(n) such that first digit = last digit: A346301;
positions in W of words w(n) such that first digit != last digit: A346302;
positions in W of words w(n) such that 1st digit = 0 and last digit 0: A171757;
positions in W of words w(n) such that 1st digit = 0 and last digit 1: A346303;
positions in W of words w(n) such that 1st digit = 1 and last digit 0: A346304;
positions in W of words w(n) such that 1st digit = 1 and last digit 1: A346305;
position in W of n-th positive integer (base 2): A206332;
positions in W of binary complement of w(n): A346306;
sum of digits in w(n): A048881;
number of runs in w(n): A346307;
positions in W of palindromes: A346308;
positions in W of words such that #0's - #1's is odd: A346309;
positions in W of words such that #0's - #1's is even: A346310;
positions in W of the reversal of the n-th word in W: A081241. (End)

Examples

			0,
1,
0,0,
0,1,
1,0,
1,1,
0,0,0,
0,0,1,
0,1,0,
0,1,1,
1,0,0,
1,0,1,
...

References

  • Bodil Branner, Dynamics, Chap. IV.14 of The Princeton Companion to Mathematics, ed. T. Gowers, p. 499.
  • K. Dajani and C. Kraaikamp, Ergodic Theory of Numbers, Math. Assoc. America, 2002, p. 72.

Links

Crossrefs

Cf. A007931, A030308, A053645.
Cf. A341256, A341258, A341334, A342753, A342910.

Programs

  • Haskell
    import Data.List (unfoldr)
a076478 n = a076478_list !! n
a076478_list = concat $ tail $ map (tail . reverse . unfoldr
   (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2 )) [1..]
-- Reinhard Zumkeller, Feb 08 2012
  • Haskell
    a076478_row n = a076478_tabf !! n :: [[Int]]
a076478_tabf = tail $ iterate (\bs -> map (0 :) bs ++ map (1 :) bs) [[]]
a076478_list' = concat $ concat a076478_tabf
-- Reinhard Zumkeller, Aug 18 2015
  • Mathematica
    d[n_] := Rest@IntegerDigits[n + 1, 2] + 1; -1 + Flatten[Array[d, 50]] (* Clark Kimberling, Feb 07 2012 *)
z = 1000;
t1 = Table[Tuples[{0, 1}, n], {n, 1, 10}];
"All binary words, lexicographic order:"
tt = Flatten[t1, 1]; (* all binary words, lexicographic order *)
"All binary words, flattened:"
Flatten[tt];
w[n_] := tt[[n]];
"List tt of all binary words:"
tt = Table[w[n], {n, 1, z}]; (*  all the binary words *)
u1 = Flatten[tt]; (* words, concatenated, A076478, binary Champernowne sequence *)
u2 = Map[Length, tt];
"Positions of 0^n:"
Flatten[Position[Map[Union, tt], {0}]]
"Positions of 1^n:"
Flatten[Position[Map[Union, tt], {1}]]
"Positions of words in which #0's = #1's:"  (* A258410 *)
"This and the next two sequences partition N."
u3 = Select[Range[Length[tt]], Count[tt[[#]], 0] == Count[tt[[#]], 1] &]
"Positions of words in which #0's < #1's:"  (* A346299 *)
u4 = Select[Range[Length[tt]], Count[tt[[#]], 0] < Count[tt[[#]], 1] &]
"Positions of words in which #0's > #1's:"  (* A346300 *)
u5 = Select[Range[Length[tt]], Count[tt[[#]], 0] > Count[tt[[#]], 1] &]
"Positions of words ending with 0:" (* A005498 *)
u6 = Select[Range[Length[tt]], Last[tt[[#]]] == 0 &]
"Positions of words ending with 1:" (* A005843 *)
u7 = Select[Range[Length[tt]], Last[tt[[#]]] == 1 &]
"Positions of words starting and ending with same digit:" (* A346301 *)
u8 = Select[Range[Length[tt]], First[tt[[#]]] == Last[tt[[#]]] &]
"Positions of words starting and ending with opposite digits:" (* A346302  *)
u9 = Select[Range[Length[tt]], First[tt[[#]]] != Last[tt[[#]]] &]
"Positions of words starting with 0 and ending with 0:" (* A346303 *)
"This and the next three sequences partition N."
u10 = Select[Range[Length[tt]], First[tt[[#]]] == 0 && Last[tt[[#]]] == 0 &]
"Positions of words starting with 0 and ending with 1:" (* A171757 *)
u11 = Select[Range[Length[tt]], First[tt[[#]]] == 0 && Last[tt[[#]]] == 1 &]
"Positions of words starting with 1 and ending with 0:" (* A346304 *)
u12 = Select[Range[Length[tt]], First[tt[[#]]] == 1 && Last[tt[[#]]] == 0 &]
"Positions of words starting with 1 and ending with 1:" (* A346305 *)
u13 = Select[Range[Length[tt]], First[tt[[#]]] == 1 && Last[tt[[#]]] == 1 &]
"Position of n-th positive integer (base 2) in tt:"
d[n_] := If[First[w[n]] == 1, FromDigits[w[n], 2]];
u14 = Flatten[Table[Position[Table[d[n], {n, 1, 200}], n], {n, 1, 200}]] (* A206332 *)
"Position of binary complement of w(n):"
u15 = comp = Flatten[Table[Position[tt, 1 - w[n]], {n, 1, 50}]] (* A346306 *)
"Sum of digits of w(n):"
u16 = Table[Total[w[n]], {n, 1, 100}] (* A048881 *)
"Number of runs in w(n):"
u17 = Map[Length, Table[Map[Length, Split[w[n]]], {n, 1, 100}]] (* A346307 *)
"Palindromes:"
Select[tt, # == Reverse[#] &]
"Positions of palindromes:"
u18 = Select[Range[Length[tt]], tt[[#]] == Reverse[tt[[#]]] &] (* A346308 *)
"Positions of words in which #0's - #1's is odd:"
u19 = Select[Range[Length[tt]], OddQ[Count[w[#], 0] - Count[w[#], 1]] &] (* A346309 *)
"Positions of words in which #0's - #1's is even:"
u20 = Select[Range[Length[tt]], EvenQ[Count[w[#], 0] - Count[w[#], 1]] &] (* A346310 *)
"Position of the reversal of the n-th word:"  (* A081241 *)
u21 = Flatten[Table[Position[tt, Reverse[w[n]]], {n, 1, 150}]]
(* Clark Kimberling, Jul 18 2011 *)
  • PARI
    {m=5; for(d=1,m, for(k=0,2^d-1,v=binary(k); while(matsize(v)[2]
  • PARI
    listn(n)= my(a=List(), i=0, s=0); while(s<=n, listput(~a, binary(i++)[^1]); s+=#a[#a]); concat(a)[1..n+1]; \\ Ruud H.G. van Tol, Mar 17 2025
  • Python
    from itertools import count, product
def agen():
    for digits in count(1):
        for b in product([0, 1], repeat=digits):
            yield from b
g = agen()
print([next(g) for n in range(105)]) # Michael S. Branicky, Jul 18 2021

Formula

To get the m-th binary vector, write m+1 in base 2 and remove the initial 1. - Clark Kimberling, Feb 07 2010

Extensions

Extended by Klaus Brockhaus, Nov 11 2002

A356088 a(n) = A001951(A022838(n)).

Original entry on oeis.org

1, 4, 7, 8, 11, 14, 16, 18, 21, 24, 26, 28, 31, 33, 35, 38, 41, 43, 45, 48, 50, 53, 55, 57, 60, 63, 65, 67, 70, 72, 74, 77, 80, 82, 84, 87, 90, 91, 94, 97, 100, 101, 104, 107, 108, 111, 114, 117, 118, 121, 124, 127, 128, 131, 134, 135, 138, 141, 144, 145
Offset: 1

Views

Author

Clark Kimberling, Aug 04 2022

Keywords

Comments

This is the first of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) u o v, defined by (u o v)(n) = u(v(n));
(2) u o v';
(3) u' o v;
(4) u' o v'.
Every positive integer is in exactly one of the four sequences.
Assume that if w is any of the sequences u, v, u', v', then lim_{n->oo} w(n)/n exists and defines the (limiting) density of w. For w = u,v,u',v', denote the densities by r,s,r',s'. Then the densities of sequences (1)-(4) exist, and 1/(r*r') + 1/(r*s') + 1/(s*s') + 1/(s*r') = 1.
For A356088, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor(n*sqrt(3)), so that r = sqrt(2), s = sqrt(3), r' = 2 + sqrt(2), s' = (3 + sqrt(3))/2.

Examples

			(1)  u o v   = (1,  4,  7,  8, 11, 14, 16, 18, 21, 24, 26, ...) = A356088.
(2)  u o v'  = (2,  5,  9, 12, 15, 19, 22, 25, 29, 32, 36, ...) = A356089.
(3)  u' o v  = (3, 10, 17, 20, 27, 34, 40, 44, 51, 58, 64, ...) = A356090.
(4)  u' o v' = (6, 13, 23, 30, 37, 47, 54, 61, 71, 78, 88, ...) = A356091.

Crossrefs

Cf. A001951, A001952, A022838, A054406, A346308 (intersections instead of results of composition), A356089, A356090, A356091.

Programs

  • Mathematica
    z = 600; zz = 100;
u = Table[Floor[n*Sqrt[2]], {n, 1, z}];  (* A001951 *)
u1 = Complement[Range[Max[u]], u];  (* A001952 *)
v = Table[Floor[n*Sqrt[3]], {n, 1, z}];  (* A022838 *)
v1 = Complement[Range[Max[v]], v];  (* A054406 *)
Table[u[[v[[n]]]], {n, 1, zz}];    (* A356088 *)
Table[u[[v1[[n]]]], {n, 1, zz}];   (* A356089 *)
Table[u1[[v[[n]]]], {n, 1, zz}];   (* A356090 *)
Table[u1[[v1[[n]]]], {n, 1, zz}];  (* A356091 *)
  • Python
    from math import isqrt
def A356088(n): return isqrt(isqrt(3*n*n)**2<<1) # Chai Wah Wu, Aug 06 2022

A351415 Intersection of Beatty sequences for (1+sqrt(5))/2 and sqrt(5).

Original entry on oeis.org

4, 6, 8, 11, 17, 22, 24, 29, 33, 35, 38, 40, 42, 46, 51, 53, 55, 58, 64, 67, 69, 71, 76, 80, 82, 84, 87, 93, 98, 100, 105, 111, 114, 116, 118, 122, 127, 129, 131, 134, 140, 145, 147, 152, 156, 158, 160, 163, 165, 169, 174, 176, 181, 187, 190, 192, 194, 199
Offset: 1

Views

Author

Clark Kimberling, Feb 10 2022

Keywords

Comments

Conjecture: every term of the difference sequence is in {2,3,4,5,6}, and each occurs infinitely many times.
From Clark Kimberling, Jul 29 2022: (Start)
This is the first of four sequences that partition the positive integers. Starting with a general overview, suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their complements, and assume that the following four sequences are infinite:
(1) u ^ v = intersection of u and v (in increasing order);
(2) u ^ v';
(3) u' ^ v;
(4) u' ^ v'.
Every positive integer is in exactly one of the four sequences. For A351415, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*(1+sqrt(5))/2) and v(n) = floor(n*sqrt(5)), so that r = (1+sqrt(5))/2, s = sqrt(5), r' = (3+sqrt(5))/2, s' = (5 + sqrt(5))/4.
(1) u ^ v = (4, 6, 8, 11, 17, 22, 24, 29, 33, 35, 38, 40, 42, ...) = A351415
(2) u ^ v' = (1, 3, 9, 12, 14, 16, 19, 21, 25, 27, 30, 32, ...) = A356101
(3) u' ^ v = (2, 13, 15, 20, 26, 31, 44, 49, 60, 62, 73, 78, ...) = A356102
(4) u' ^ v' = (5, 7, 10, 18, 23, 28, 34, 36, 39, 41, 47, 52, ...) = A356103
(End)

Examples

			The two Beatty sequences are (1,3,4,6,8,9,11,12,14,...) and (2,4,6,8,11,13,15,17,...), with common terms forming the sequence (4,6,8,11,...).

Crossrefs

Cf. A000201, A022839, A296184, A346308, A347469, A347793.
Cf. A001950, A108598, A356101, A356102, A356103, A356104 (results of composition instead of intersections), A190509 (composites, reversed order).

Programs

  • Mathematica
    z = 200;
r = (1 + Sqrt[5])/2; u = Table[Floor[n*r], {n, 1, z}]  (* A000201 *)
u1 = Take[Complement[Range[1000], u], z]  (* A001950 *)
r1 = Sqrt[5]; v = Table[Floor[n*r1], {n, 1, z}]  (* A022839 *)
v1 = Take[Complement[Range[1000], v], z]  (* A108598 *)
Intersection[u, v]    (* A351415 *)
Intersection[u, v1]   (* A356101 *)
Intersection[u1, v]   (* A356102 *)
Intersection[u1, v1]  (* A356103 *)

A347469 For irrational r > 1, let B(r) denote the Beatty sequence for r. Let (s(n)) be the increasing sequence of numbers in both B(sqrt(2)) and B(sqrt(3)). Then this sequence gives the numbers k such that s(k) = floor(k*sqrt(6)).

Original entry on oeis.org

1, 20, 21, 23, 49, 70, 71, 72, 73, 74, 75, 76, 78, 98, 101, 102, 117, 148, 194, 215, 216, 250, 257, 262, 299, 300, 307, 310, 344, 346, 357, 360, 361, 448, 1071, 1075, 1083, 1114, 1143, 1160, 1203, 1681, 1722, 1725, 1727, 1737, 1740, 1741, 1770, 1771, 1783
Offset: 1

Views

Author

Clark Kimberling, Oct 31 2021

Keywords

Comments

It is conjectured that this sequence is infinite.

Examples

			(B(sqrt(2)) and B(sqrt(3))) - B(sqrt(6)) = (0, -1, 1, 1, 3, 3, 5, 5, 5, 3, 5, 5, 3, 5, 4, 3, 2, 2, 1, 0, 0, -1, 0, -1, -1, -1, -1, -1, -1, -2, ...), so that a(4) = 23, the position of the 4th 0.

Crossrefs

Cf. A001951, A022838, A022840, A346308, A347467, A347468.

Programs

  • Mathematica
    z = 10000; r = Sqrt[2]; s = Sqrt[3];
u = Table[Floor[n r], {n, 0, z}];  (*A001951*)
v = Table[Floor[n s], {n, 1, z}];  (*A022838*)
w = Intersection[u, v];  (*A346308*)
zz = -1 + Length[w];
t = Table[Floor[n*r*s], {n, 0, zz}]; (* A022840 *)
d = w - t;
Flatten[Position[d, 0]] (* A347469 *)

A356091 a(n) = A001952(A054406(n)).

Original entry on oeis.org

6, 13, 23, 30, 37, 47, 54, 61, 71, 78, 88, 95, 102, 112, 119, 126, 136, 143, 150, 160, 167, 177, 184, 191, 201, 208, 215, 225, 232, 238, 249, 256, 266, 273, 279, 290, 297, 303, 314, 320, 331, 338, 344, 355, 361, 368, 378, 385, 392, 402, 409, 419, 426, 433
Offset: 1

Views

Author

Clark Kimberling, Aug 05 2022

Keywords

Comments

This is the fourth of four sequences that partition the positive integers. See A356088.

Examples

			(1)  u o v   = (1,  4,  7,  8, 11, 14, 16, 18, 21, 24, 26, ...) = A356088
(2)  u o v'  = (2,  5,  9, 12, 15, 19, 22, 25, 29, 32, 36, ...) = A356089
(3)  u' o v  = (3, 10, 17, 20, 27, 34, 40, 44, 51, 58, 64, ...) = A356090
(4)  u' o v' = (6, 13, 23, 30, 37, 47, 54, 61, 71, 78, 88, ...) = A356091

Crossrefs

Cf. A001951, A001952, A022838, A054406, A346308 (intersections instead of results of composition), A356088, A356089, A356090.

Programs

  • Mathematica
    z = 600; zz = 100;
u = Table[Floor[n*Sqrt[2]], {n, 1, z}];  (* A001951 *)
u1 = Complement[Range[Max[u]], u];  (* A001952 *)
v = Table[Floor[n*Sqrt[3]], {n, 1, z}];  (* A022838 *)
v1 = Complement[Range[Max[v]], v];  (* A054406 *)
Table[u[[v[[n]]]], {n, 1, zz}]    (* A356088 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A356089 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* A356090 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A356091 *)

A347467 Numbers h such that floor(k*sqrt(3)) = floor(h*sqrt(2)) for some k.

Original entry on oeis.org

1, 4, 6, 9, 11, 14, 16, 17, 18, 21, 22, 23, 26, 27, 28, 29, 31, 32, 33, 34, 36, 38, 39, 41, 43, 44, 46, 48, 49, 51, 53, 54, 55, 56, 59, 60, 61, 64, 65, 66, 68, 70, 71, 73, 76, 78, 81, 83, 86, 88, 91, 93, 96, 98, 99, 101, 103, 104, 105, 108, 109, 110, 113
Offset: 1

Views

Author

Clark Kimberling, Oct 16 2021

Keywords

Examples

			Beatty sequence for sqrt(2): (1,2,4,5,7,8,9,11,12,14,...)
Beatty sequence for sqrt(3): (1,3,5,6,8,10,12,13,15,...)
Intersection: (1,5,8,12,...), as in A346308.
a(2) = 4 because floor(3*sqrt(3)) = floor(4*sqrt(2)).  (For each such h, there is only one such k.)

Crossrefs

Cf. A001951, A022838, A346308, A347468, A347469.

Programs

  • Mathematica
    z = 200; r = Sqrt[2]; s = Sqrt[3];
u = Table[Floor[n r], {n, 0, z}]; (*A001951*)
v = Table[Floor[n s], {n, 1, z}]; (*A022838*)
w = Intersection[u, v]  (*A346308*)
zz = -1 + Length[w];
Table[Ceiling[w[[n]]/r], {n, 1, zz}] (* A347467 *)
Table[Ceiling[w[[n]]/s], {n, 1, zz}] (* A347468 *)

A347468 Numbers k such that floor(k*sqrt(3)) = floor(h*sqrt(2)) for some h.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 35, 36, 38, 39, 40, 42, 43, 44, 45, 46, 48, 49, 50, 52, 53, 54, 56, 57, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 81, 82, 84, 85, 86, 88, 89, 90, 92, 93, 94, 95
Offset: 1

Views

Author

Clark Kimberling, Oct 28 2021

Keywords

Examples

			Beatty sequence for sqrt(2): (1,2,4,5,7,8,9,11,12,14,...)
Beatty sequence for sqrt(3): (1,3,5,6,8,10,12,13,15,...)
Intersection: (1,5,8,12,...), as in A346308.
a(2) = 3 because floor(3*sqrt(3)) = floor(4*sqrt(2)).  (For each such k, there is only one such h.)

Crossrefs

Cf. A001951, A022838, A346308, A347467, A347469.

Programs

  • Mathematica
    z = 200; r = Sqrt[2]; s = Sqrt[3];
u = Table[Floor[n r], {n, 0, z}]; (*A001951*)
v = Table[Floor[n s], {n, 1, z}]; (*A022838*)
w = Intersection[u, v]  (*A346308*)
zz = -1 + Length[w];
Table[Ceiling[w[[n]]/r], {n, 1, zz}] (* A347467 *)
Table[Ceiling[w[[n]]/s], {n, 1, zz}] (* A347468 *)

A356085 Intersection of A001951 and A054406.

Original entry on oeis.org

2, 4, 7, 9, 11, 14, 16, 18, 21, 26, 28, 33, 35, 42, 49, 52, 56, 59, 63, 66, 70, 73, 80, 82, 87, 89, 94, 97, 101, 104, 106, 108, 111, 113, 115, 118, 120, 123, 125, 127, 130, 132, 134, 137, 141, 144, 149, 151, 156, 158, 165, 172, 175, 179, 182, 186, 189, 196
Offset: 1

Views

Author

Clark Kimberling, Jul 26 2022

Keywords

Comments

This is the second of four sequences, u^v, u^v', u'^v, u'^v', that partition the positive integers. See A346308.

Examples

			(1)  u ^ v = (1, 5, 8, 12, 15, 19, 22, 24, 25, 29, 31, 32, ...) =  A346308
(2)  u ^ v' = (2, 4, 7, 9, 11, 14, 16, 18, 21, 26, 28, 33, 35, ...) =  A356085
(3)  u' ^ v = (3, 6, 10, 13, 17, 20, 27, 34, 51, 58, 64, 71, 81, ...) = A356086
(4)  u' ^ v' = (23, 30, 37, 40, 44, 47, 54, 61, 68, 75, 78, 85, ...) = A356087

Crossrefs

Cf. A001951, A001952, A022838, A054406, A346308, A356086, A356087, A356088 (composites instead of intersections).

Programs

  • Mathematica
    z = 200;
r = Sqrt[2]; u = Table[Floor[n*r], {n, 1, z}]  (* A001951 *)
u1 = Take[Complement[Range[1000], u], z]  (* A001952 *)
r1 = Sqrt[3]; v = Table[Floor[n*r1], {n, 1, z}]  (* A022838 *)
v1 = Take[Complement[Range[1000], v], z]  (* A054406 *)
t1 = Intersection[u, v]    (* A346308 *)
t2 = Intersection[u, v1]   (* A356085 *)
t3 = Intersection[u1, v]   (* A356086 *)
t4 = Intersection[u1, v1]  (* A356087 *)

A356086 Intersection of A001952 and A022838.

Original entry on oeis.org

3, 6, 10, 13, 17, 20, 27, 34, 51, 58, 64, 71, 81, 88, 95, 102, 105, 109, 112, 116, 119, 122, 126, 129, 133, 136, 143, 150, 157, 174, 180, 187, 204, 211, 218, 221, 225, 228, 232, 235, 242, 245, 249, 252, 256, 259, 266, 273, 284, 285, 287, 289, 290, 292, 294
Offset: 1

Views

Author

Clark Kimberling, Aug 04 2022

Keywords

Comments

This is the third of four sequences, u^v, u^v', u'^v, u'^v', that partition the positive integers. See A346308.

Examples

			(1)  u ^ v   = ( 1,  5,  8, 12, 15, 19, 22, 24, 25, 29, 31, 32, ...) = A346308
(2)  u ^ v'  = ( 2,  4,  7,  9, 11, 14, 16, 18, 21, 26, 28, 33, ...) = A356085
(3)  u' ^ v  = ( 3,  6, 10, 13, 17, 20, 27, 34, 51, 58, 64, 71, ...) = A356086
(4)  u' ^ v' = (23, 30, 37, 40, 44, 47, 54, 61, 68, 75, 78, 85, ...) = A356087

Crossrefs

Cf. A001951, A001952, A022838, A054406, A346308, A356085, A356087, A356088 (results of composition instead of intersections).

Programs

  • Mathematica
    z = 200;
r = Sqrt[2]; u = Table[Floor[n*r], {n, 1, z}]  (* A001951 *)
u1 = Take[Complement[Range[1000], u], z]  (* A001952 *)
r1 = Sqrt[3]; v = Table[Floor[n*r1], {n, 1, z}]  (* A022838 *)
v1 = Take[Complement[Range[1000], v], z]  (* A054406 *)
Intersection[u, v]    (* A346308 *)
Intersection[u, v1]   (* A356085 *)
Intersection[u1, v]   (* A356086 *)
Intersection[u1, v1]  (* A356087 *)
  • Python
    from math import isqrt
from itertools import count, islice
def A356086_gen(): # generator of terms
    return filter(lambda n:n == isqrt(3*(isqrt(n**2//3)+1)**2),((k:=n<<1)+isqrt(k*n) for n in count(1)))
A356086_list = list(islice(A356086_gen(),30)) # Chai Wah Wu, Aug 06 2022

A356087 Intersection of A001952 and A054406.

Original entry on oeis.org

23, 30, 37, 40, 44, 47, 54, 61, 68, 75, 78, 85, 92, 99, 139, 146, 153, 160, 163, 167, 170, 177, 184, 191, 194, 198, 201, 208, 215, 238, 262, 269, 276, 279, 283, 286, 288, 291, 293, 295, 298, 300, 302, 305, 307, 309, 312, 314, 317, 319, 321, 324, 326, 328
Offset: 1

Views

Author

Clark Kimberling, Aug 04 2022

Keywords

Comments

This is the fourth of four sequences, u^v, u^v', u'^v, u'^v', that partition the positive integers. See A346308.

Examples

			(1)  u ^ v   = ( 1,  5,  8, 12, 15, 19, 22, 24, 25, 29, 31, 32, ...) = A346308.
(2)  u ^ v'  = ( 2,  4,  7,  9, 11, 14, 16, 18, 21, 26, 28, 33, ...) = A356085.
(3)  u' ^ v  = ( 3,  6, 10, 13, 17, 20, 27, 34, 51, 58, 64, 71, ...) = A356086.
(4)  u' ^ v' = (23, 30, 37, 40, 44, 47, 54, 61, 68, 75, 78, 85, ...) = A356087.

Crossrefs

Cf. A001951, A001952, A022838, A054406, A346308, A356085, A356086, A356088 (results of composition instead of intersections).

Programs

  • Mathematica
    r = Sqrt[2]; u = Table[Floor[n*r], {n, 1, z}]  (* A001951 *)
u1 = Take[Complement[Range[1000], u], z]  (* A001952 *)
r1 = Sqrt[3]; v = Table[Floor[n*r1], {n, 1, z}]  (* A022838 *)
v1 = Take[Complement[Range[1000], v], z]  (* A054406 *)
Intersection[u, v]    (* A346308 *)
Intersection[u, v1]   (* A356085 *)
Intersection[u1, v]   (* A356086 *)
Intersection[u1, v1]  (* A356087 *)
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